Reverse Recovery Time is the time it takes a diode to reduce the reverse current to zero when going from forward bias to reverse bias. Contrary to popular opinion and idealized models diodes do conduct in the reverse direction, but only for a short while.

A fast recovery diode has a smaller reverse recovery time than a standard diode.

I'm not familiar with the term soft recovery, but it sounds like it should be slower and less abrupt.

The reason for reverse recovery is that a diode is a minority carrier device. It takes a certain amount of time for them to recombine.

Fast recovery diodes were designed for use in switching applications such as switching power supplies to reduce the recovery time. If you use a standard recovery diode in a switching power supply, the diode will cause enormous inefficiencies.

However, it was later realized that having a very fast recovery time caused other issues with switching power supplies. I can't remember exactly what it was, but it had something to do with its interaction with the switching FET. To fix this, soft recovery diodes were developed. They recover quickly but with a softer return to zero current.